Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure

Peña, Daniel; Prieto, Francisco J.; Viladomat, Júlia

doi:10.1016/j.jmva.2010.04.014

Cited by 34 publications

(18 citation statements)

References 22 publications

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“…The first way is through numerical optimization and the second finds the eigenvectors of some generalized kurtosis matrix. In Peña et al (2008) we have compared these two approaches in a particular case. Given a multivariate random vector X with mean μ and covariance matrix Σ, we propose to compute the eigenvectors of the kurtosis matrix K = E ( Z T ZZZ T ), where Z =Σ −1/2 ( X − μ ).…”

Section: Discussion On the Paper By Tyler Critchley Dümbgen And Ojamentioning

confidence: 99%

Invariant Co-Ordinate Selection

Tyler

Critchley

Dümbgen

et al. 2009

Journal of the Royal Statistical Society Series B: Statistical Methodology

115

132

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Summary.A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue-eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for invariant co-ordinate selection. By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant coordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.

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Section: Discussion On the Paper By Tyler Critchley Dümbgen And Ojamentioning

confidence: 99%

Invariant Co-Ordinate Selection

Tyler

Critchley

Dümbgen

et al. 2009

Journal of the Royal Statistical Society Series B: Statistical Methodology

115

132

View full text Add to dashboard Cite

show abstract

“…Due to the fact that Projection Pursuit methods execution time increases exponentially with the dimensionality of data, these algorithms tend to be computationally intensive. This motivated researches to find alternative approaches for pursuing interesting projection not only in terms of data structures, but also in terms of computational resources [25,26]. For example, in [27] a simple version of the Projection Pursuit-based estimator is proposed which is easy to implement and fast to compute.…”

Section: Projection Pursuit Algorithmsmentioning

confidence: 99%

Accelerating a Geometrical Approximated PCA Algorithm Using AVX2 and CUDA

Machidon

Ciobanu

et al. 2020

Remote Sensing

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Remote sensing data has known an explosive growth in the past decade. This has led to the need for efficient dimensionality reduction techniques, mathematical procedures that transform the high-dimensional data into a meaningful, reduced representation. Projection Pursuit (PP) based algorithms were shown to be efficient solutions for performing dimensionality reduction on large datasets by searching low-dimensional projections of the data where meaningful structures are exposed. However, PP faces computational difficulties in dealing with very large datasets—which are common in hyperspectral imaging, thus raising the challenge for implementing such algorithms using the latest High Performance Computing approaches. In this paper, a PP-based geometrical approximated Principal Component Analysis algorithm (gaPCA) for hyperspectral image analysis is implemented and assessed on multi-core Central Processing Units (CPUs), Graphics Processing Units (GPUs) and multi-core CPUs using Single Instruction, Multiple Data (SIMD) AVX2 (Advanced Vector eXtensions) intrinsics, which provide significant improvements in performance and energy usage over the single-core implementation. Thus, this paper presents a cross-platform and cross-language perspective, having several implementations of the gaPCA algorithm in Matlab, Python, C++ and GPU implementations based on NVIDIA Compute Unified Device Architecture (CUDA). The evaluation of the proposed solutions is performed with respect to the execution time and energy consumption. The experimental evaluation has shown not only the advantage of using CUDA programming in implementing the gaPCA algorithm on a GPU in terms of performance and energy consumption, but also significant benefits in implementing it on the multi-core CPU using AVX2 intrinsics.

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“…Hence, ICS is often considered as a preprocessing step for clustering or outlier detection. The interested reader is referred to Alashwali and Kent (); Archimbaud, Nordhausen, and Ruiz‐Gazen (); Bugrien and Kent (); Pena and Prieto (); Pena, Prieto, and Viladomat (); Tyler et al (); and references therein for further details. As a final comment, note also that many supervised dimension reduction methods like SIR (K.‐C.…”

Section: Classical Ica Solutions: Fobi and Jadementioning

confidence: 99%

Independent component analysis: A statistical perspective

Nordhausen¹,

Oja

2018

WIREs Computational Stats

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1. z $ N p (0, I p ) ) multivariate normal (Gaussian) model. 2. z spherical, that is, Uz $ z for all orthogonal U ) elliptical model. 3. z has independent components ) independent component model. Both the elliptical model and IC model are extensions of the Gaussian model but in diverse directions. In the case of the Gaussian and elliptical model, A is identifiable only up to a postmultiplication by an orthogonal matrix, and the inference is only for parameters E(x) = b and Cov(x) = AA 0. Classical multivariate statistical inference methods, such as Hotelling's T 2 , multivariate analysis of variance, multivariate regression, and PCA, rely on the assumption of multivariate normality. Gaussianity and ellipticity assumptions mean that no hidden structures or latent variables are allowed in the data analysis.The IC model is a so called BSS model with latent variables in z and latent structures caused by different marginal distributions of the components of z. In the case of the IC model one tries to estimate an unmixing matrix to go back to the non-2 of 23 NORDHAUSEN AND OJA

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Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure

Cited by 34 publications

References 22 publications

Invariant Co-Ordinate Selection

Invariant Co-Ordinate Selection

Accelerating a Geometrical Approximated PCA Algorithm Using AVX2 and CUDA

Independent component analysis: A statistical perspective

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